Learner Guide. Cambridge International AS & A Level Mathematics

Transcription

1 Learner Guide Cambridge International AS & A Level Mathematics 9709

2 Cambridge International Examinations retains the copyright on all its publications. Registered Centres are permitted to copy material from this booklet for their own internal use. However, we cannot give permission to Centres to photocopy any material that is acknowledged to a third party even for internal use within a Centre. Cambridge International Examinations 2015 Version 2.1 Updated:

3 Contents How to use this guide... 3 Section 1: How will you be tested? Section 2: Examination advice Section 3: What will be tested? Section 4: What you need to know Section 5: Useful websites Section 1: How will you be tested?... 5 About the examination papers About the papers Section 2: Examination advice... 7 How to use this advice General advice Advice for P1, P2 and P3 Advice for M1 and M2 Advice for S1 and S2 Section 3: What will be tested? Section 4: What you need to know How to use the table Pure Mathematics 1: Unit P1 Pure Mathematics 2: Unit P2 Pure Mathematics 3: Unit P3 Mechanics 1: Unit M1 Mechanics 2: Unit M2 Probability and Statistics 1: Unit S1 Probability and Statistics 2: Unit S2 Section 5: Useful websites...34

4 2 Cambridge International AS and A Level Mathematics 9709

5 How to use this guide How to use this guide This guide describes what you need to know about your Cambridge International AS and A Level Mathematics examination. It will help you plan your revision programme and it will explain what we are looking for in your answers. It can also be used to help you revise, by using the topic lists in Section 4 to check what you know and which topic areas you have covered. The guide contains the following sections. Section 1: How will you be tested? This section will give you information about the different examination papers that are available. Section 2: Examination advice This section gives you advice to help you do as well as you can. Some of the ideas are general advice and some are based on the common mistakes that learners make in exams. Section 3: What will be tested? This section describes the areas of knowledge, understanding and skills that you will be tested on. Section 4: What you need to know This section shows the syllabus content in a simple way so that you can check: what you need to know about each topic; how much of the syllabus you have covered. Not all the information in this checklist will be relevant to you. You will need to select what you need to know in Sections 1, 2 and 4 by finding out from your teacher which examination papers you are taking. Section 5: Useful websites Cambridge International AS and A Level Mathematics

6 How to use this guide 4 Cambridge International AS and A Level Mathematics 9709

7 Section 1: How will you be tested? Section 1: How will you be tested? About the examination papers This syllabus is made up of seven units: three of them are Pure Mathematics units (P1, P2 and P3), two are Mechanics units (M1 and M2) and two are Probability and Statistics units (S1 and S2). Each unit has its own examination paper. You need to take two units for the Cambridge International AS Level qualification or four units for the full Cambridge International A Level qualification. The table below shows which units are compulsory and which are optional, and shows also the weighting of each unit towards the overall qualification. You will need to check with your teacher which units you will be taking. Certification title Compulsory units Optional units Cambridge International AS Level Mathematics Cambridge International A Level Mathematics P1 (60%) P1 (30%) and P3 (30%) P2 (40%) or M1 (40%) or S1 (40%) M1 (20%) and S1 (20%) or M1 (20%) and M2 (20%) or S1 (20%) and S2 (20%) The following combinations of units are possible. AS Level P1 and P2 P1 and M1 P1 and S1 A Level P1, P3, M1 and S1 P1, P3, M1 and M2 P1, P3, S1 and S2 If you are taking the full A Level qualification, you can take the examination papers for all four units in one session, or alternatively you can take two of them (P1 and M1 or P1 and S1) at an earlier session for an AS qualification and then take your other two units later. The AS Level combination of units P1 and P2 cannot be used as the first half of a full A Level qualification, so you should not be taking P2 if you intend to do the full Cambridge International A Level Mathematics qualification. Cambridge International AS and A Level Mathematics

8 Section 1: How will you be tested? About the papers Once you have checked with your teacher which units you are doing you can use the table below to find basic information about each unit. All units are assessed by a written examination, externally set and marked. You must answer all questions and all relevant working must be clearly shown. Each paper will contain both shorter and longer questions, with the questions being arranged approximately in order of increasing mark allocation (i.e. questions with smaller numbers of marks will come earlier in the paper and those with larger numbers of marks will come later in the paper.) Units P1 and P3 will contain about 10 questions and the other units about 7 questions. Component Unit name Total marks Duration Qualification use Paper 1 Paper 2 Paper 3 Paper 4 Paper 5 Paper 6 Paper 7 P1 Pure Mathematics 1 P2 Pure Mathematics 2 P3 Pure Mathematics 3 M1 Mechanics 1 M2 Mechanics 2 S1 Probability and Statistics 1 S2 Probability and Statistics hour 45 mins AS Level Mathematics A Level Mathematics 50 1 hour 15 mins AS Level Mathematics 75 1 hour 45 mins A Level Mathematics 50 1 hour 15 mins AS Level Mathematics A Level Mathematics 50 1 hour 15 mins A Level Mathematics 50 1 hour 15 mins AS Level Mathematics A Level Mathematics 50 1 hour 15 mins A Level Mathematics 6 Cambridge International AS and A Level Mathematics 9709

9 Section 2: Examination advice Section 2: Examination advice How to use this advice This section highlights some common mistakes made by learners. They are collected under various subheadings to help you when you revise a particular topic or area. General advice You should give numerical answers correct to three significant figures in questions where no accuracy is specified, except for angles in degrees when one decimal place accuracy is required. To achieve three-figure accuracy in your answer you will have to work with at least four figures in your working throughout the question. For a calculation with several stages, it is usually best to use all the figures that your calculator shows; however you do not need to write all these figures down in your working. Giving too many significant figures in an answer is not usually penalised. However, if the question does specify an accuracy level then you must keep to it for your final answer, and giving too many significant figures here will be penalised. There are some questions which ask for answers in exact form. In these questions you must not use your calculator to evaluate answers. Exact answers may include fractions, square roots and constants such as p, for example, and you should give them in as simple a form as possible. You are expected to use a scientific calculator in all examination papers. Computers, graphical calculators and calculators capable of algebraic manipulation are not permitted. Make sure you check that your calculator is in degree mode for questions that have angles in degrees, and in radian mode when you require or are given angles in radians. You should always show your working, as marks are usually awarded for using a correct method even if you make a mistake somewhere. It is particularly important to show all your working in a question where there is a given answer. Marks will not be gained if the examiner is not convinced of the steps in your working out. Read questions carefully. Misreading a question can cost you marks. Write clearly. Make sure all numbers are clear, for example make sure your 1 s don t look like 7 s. If you need to change a word or a number, or even a sign (+ to for example) it is best to cross out and re-write it. Don t try to write over the top of your previous work. If your alteration is not clear you will not get the marks. When an answer is given (i.e. the question says show that ), it is often because the answer is needed in the next part of the question. So a given answer may mean that you can carry on with a question even if you haven t been able to obtain the answer yourself. When you need to use a given answer in a later part of a question, you should always use the result exactly as given in the question, even if you have obtained a different answer yourself. You should never continue with your own wrong answer, as this will lead to a further loss of marks. Make sure you are familiar with all the standard mathematical notation that is expected for this syllabus. Your teacher will be able to advise you on what is expected. Cambridge International AS and A Level Mathematics

10 Section 2: Examination advice Although there is no choice of questions in any of the Cambridge International AS and A Level papers, you do not have to answer the questions in the order they are printed in the question paper. You don t want to spend time struggling on one question and then not have time for a question that you could have done and gained marks for. Check the number of marks for each question/part question. This gives an indication of how long you should be spending. Part questions with only one or two marks should not involve you doing complicated calculations or writing long explanations. You don t want to spend too long on some questions and then run out of time at the end. A rough guide is that a rate of a mark a minute would leave you a good amount of time at the end for checking your work and for trying to complete any parts of questions that you hadn t been able to do at first. As long as you are not running out of time, don t give up on a question just because your working or answers are starting to look wrong. There are always marks available for the method used. So even if you have made a mistake somewhere you could still gain method marks. Don t cross out anything until you have replaced it even if you know it s not correct there may still be method marks to gain. There are no marks available for just stating a method or a formula. The method has to be applied to the particular question, or the formula used by substituting values in. Always look to see if your answer is reasonable. For example if you had a probability answer of 1.2, you would know that you had made a mistake and you would need to go back and check your solution. Make sure that you have answered everything that a question asks. Sometimes one sentence asks two things (e.g. Show that and hence find ) and it is easy to concentrate only on the first request and forget about the second one. Check the formula book before the examination. You must be aware which formulas are given and which ones you will need to learn. Make sure you practise lots of past examination papers so that you know the format and the type of questions. You could also time yourself when doing them so that you can judge how quickly you will need to work in the actual examination. Presentation of your work is important don t present your solutions in the examination in double column format. Make sure you know the difference between three significant figures and three decimal places, and make sure you round answers rather than truncate them. Advice for P1, P2 and P3 Make sure you know all the formulas that you need (even ones from Cambridge IGCSE). If you use an incorrect formula you will score no marks. Check to see if your answer is required in exact form. If this is the case in a trigonometry question exact values of sin 60, etc. will need to be used. Make sure you know the exact form for sin 60, etc. They are not in the formula booklet. Additional advice for P2 and P3 Calculus questions involving trigonometric functions use values in radians, so make sure your calculator is in the correct mode if you need to use it. 8 Cambridge International AS and A Level Mathematics 9709

11 Section 2: Examination advice Advice for M1 and M2 When using a numerical value for g you must use 10 m s 2 (unless the question states otherwise). Always draw clear force diagrams when appropriate, whether the question asks for them or not. Make sure you are familiar with common words and phrases such as initial, resultant, and know the difference between mass and weight. Go through some past examination papers and highlight common words and phrases. Make sure you know what they mean. Advice for S1 and S2 Questions that ask for probabilities must have answers between 0 and 1. If you get an answer greater than 1 for a probability you should check your working to try to find the mistake. Answers for probabilities can often be given as either fractions or decimals. For answers in decimal form, it is important that you know the difference between three significant figures and three decimal places. For example to three significant figures is , but to three decimal places is A final answer of would not score the accuracy mark as it is not correct to the level of accuracy required. Sometimes you may be asked to answer in the context of the question. This means that you cannot just give a textbook definition that could apply to any situation; your answer must relate to the situation given in the question. So, for example, you should not just say The events must be independent, but The scores when the die is thrown must be independent or The times taken by the people must be independent of each other or whatever it is that the question is about. When answering a question about a normal distribution, it is useful to draw a diagram. This can prevent you from making an error; e.g. if you are finding a probability, a diagram will indicate whether your answer should be greater than or less than 0.5. Additional advice for S2 When carrying out a hypothesis test, you should always state the conclusion in the context of the question. You should not state conclusions in a way that implies that a hypothesis test has proved something; e.g. it is better to say There is evidence that the mean weight of the fruit has increased than The test shows that the mean weight of the fruit has increased. When calculating a confidence interval, it may not always be sensible to apply the usual three significant figure rule about accuracy. For example, if an interval for a population mean turns out to be (99.974, ), it makes no sense to round each end to 100, and you should give two or three decimal places in a case like this, so that the width of the interval is shown with reasonable accuracy. Cambridge International AS and A Level Mathematics

12 Section 2: Examination advice 10 Cambridge International AS and A Level Mathematics 9709

13 Section 3: What will be tested? Section 3: What will be tested? The full syllabus, which your teacher will have, says that the assessment covers technique with application. This means that you will be tested both on your knowledge of all the mathematical topics, methods, etc. that are set out in the syllabus content, and also on your ability to apply your knowledge in solving problems. The syllabus also lists five assessment objectives that set out the kinds of thing that examination questions will be designed to test. These assessment objectives indicate that: you need to be familiar with all the usual notation and terminology used in your units, as well understanding the mathematical ideas you need to work accurately, e.g. in solving equations, etc. you may have to decide for yourself how to go about solving a problem, without being told exactly what procedures you need to use you may need to use more than one aspect of your knowledge to solve a problem your work needs to be clearly and logically set out. You should ask your teacher if you require more detailed information on this section. Cambridge International AS and A Level Mathematics

14 Section 3: What will be tested? 12 Cambridge International AS and A Level Mathematics 9709

15 Section 4: What you need to know This section has a table for each of the seven units. Each table lists the things you may be tested on in the examination for that unit. How to use the table You can use the table throughout your mathematics course to check the topic areas you have covered. You can also use it as a revision aid. You can go through the list at various stages in the course, and put: a RED dot against a topic if you are really unsure an ORANGE dot if you are fairly sure of the topic, but need some further practice a GREEN dot if you are fully confident. As you progress through the course and prepare for the examination you should be giving yourself more and more green dots! Remember that we are looking for you to know certain skills and facts and to be able to apply these in different situations. It is therefore important to learn the facts and practise the skills first, in isolation; but then you need to find more complex questions that use the skills you are learning, so that you can practise applying what you have learnt to solving problems. Working through past examination papers is invaluable here. You will see that the syllabus has been divided into skills, knowledge, and application. However, you should remember that each skill and knowledge item could be tested in questions requiring use of that particular skill or knowledge in a situation that may not be specifically mentioned here. Cambridge International AS and A Level Mathematics

16 Pure Mathematics 1: Unit P1 Topic Skill Knowledge Application Checklist Quadratics Complete the square Locate vertex of graph Find discriminant Sketch graph of quadratic Solve quadratic equations Solve linear and quadratic inequalities Solve by substitution a pair of simultaneous equations one linear and one quadratic Use in connection with the number of real roots Recognise and solve equations that can be put into quadratic form Solve problems, e.g. involving the intersection of a line and a curve Functions Find range of a given function Find fg(x) for given f and g Understand terms: function, domain and range Sketch graphs of a one-one function and its inverse Identify one-one functions Find inverse of one-one function one-one function, inverse function composition of functions Restrict a domain to ensure that a function is one-one Understand how the graphs of a one-one function and its inverse are related Coordinate geometry Given end points of a line: find the length find the gradient find the mid-point Find equation of a line, e.g. using 2 points on it, or 1 point and the gradient Gradients of parallel lines are equal Perpendicular gradients: m 1 m 2 = 1 Know the forms y = mx + c and y y 1 = m(x x 1 ) Understand the relationship between a graph and its equation Use these relationships in solving problems Interpret and use linear equations Solve problems that relate points of intersection of graphs to solution of equations (including the correspondence between a line being a tangent to curve and an equation having a repeated root) 14 Cambridge International AS and A Level Mathematics 9709

17 Pure Mathematics 1: Unit P1 Topic Skill Knowledge Application Checklist Circular measure Convert between radians and degrees Definition of a radian Solve problems involving arc lengths and areas of sectors and segments Use formulae s = r θ and A = ½r 2 θ to calculate arc length and sector area Trigonometry Sketch and recognise graphs of the sine, cosine and tangent functions (angles of any size, degrees or radians) Solve trig equations giving all solutions in specified interval Know exact values for sin, cos, tan of 30º, 45º, 60º Understand notation sin 1 x, cos 1 x, tan 1 x as denoting principal values Identities: sin cos i i = tan i sin 2 θ + cos 2 θ = 1 Use trigonometric graphs Use exact values of related angles e.g. cos 150º Use these identities, e.g. to prove other identities or to solve equations Vectors Add and subtract vectors, and multiply a vector by a scalar Find magnitude of a vector and the scalar product of two vectors Standard vector notations Meaning of terms: unit vector displacement vector Geometrical interpretation Use scalar products to find angles and to solve problems involving perpendicular lines position vector Series Expand (a + b) n for positive integer n Find: nth term of AP sum of AP nth term GP sum of GP Notation n! and c n m r Recognise arithmetic progressions and geometric progressions Condition for convergence of geometric progression Use arithmetic progression and/or geometric progression formulae in solving problems sum to infinity of convergent GP Cambridge International AS and A Level Mathematics

18 Pure Mathematics 1: Unit P1 Topic Skill Knowledge Application Checklist Differentiation Differentiate x n for rational n, together with constant multiples, sums and differences of functions Understand gradient of a curve 2 d y d y Notation,, ( ), ( ) d x 2 f l x f m x d x Apply differentiation to: gradients tangents and normals Use the chain rule on composite functions increasing/decreasing functions connected rates of change Find stationary points and identify maximum/minimum Use information about stationary points to sketch graphs Integration Integrate (ax + b) n (rational n 1) together with constant multiples, sums and differences. Integration as reverse differentiation Use integration to solve problems involving finding a constant of integration Evaluate definite integrals Solve problems involving: area under a curve area between two curves volume of revolution about one of the axes 16 Cambridge International AS and A Level Mathematics 9709

19 Pure Mathematics 2: Unit P2 Topic Skill Knowledge Application Checklist Assumed knowledge Content of unit P1 is assumed, and may be required in solving problems on P2 topics Algebra Solve modulus equations and inequalities, including use of: a = b a 2 = b 2 Meaning of x Meaning of quotient and remainder Use of theorems in finding factors, solving polynomial equations, finding unknown coefficients etc. x a < b a b < x < a + b Factor and remainder theorems Carry out algebraic polynomial division Logarithmic and exponential functions Solve equations of form a x = b and corresponding inequalities Relationship between logarithms and indices Laws of logarithms Use of laws, e.g. in solving equations Use graphs of e x and ln x Definition and properties of e x and ln x Transformation to linear form, and use of gradient and intercept Cambridge International AS and A Level Mathematics

20 Pure Mathematics 2: Unit P2 Topic Skill Knowledge Application Checklist Trigonometry The sec, cosec and cot functions and their relationship to cos, sin and tan Use properties and graphs of all six trig functions for angles of any magnitude Identities: sec 2 θ = 1 + tan 2 θ cosec 2 θ = 1 + cot 2 θ Expansions of: sin(a ± B) Use of these in evaluating and simplifying expressions, and in solving equations, including expressing a sin θ + b cos θ in the forms R sin(θ ± α) and R cos (θ ± α) cos(a ± B) tan(a ± B) Formulae for: sin 2A cos 2A tan 2A Differentiation Differentiate e x and ln x, sin x, cos x and tan x, together with constant multiples, sums, differences and composites Applications of differentiation include all those in unit P1 Differentiate products and quotients Parametric differentiation Implicit differentiation 18 Cambridge International AS and A Level Mathematics 9709

21 Pure Mathematics 2: Unit P2 Topic Skill Knowledge Application Checklist Integration Integrate: eax + b Carry out integrations using appropriate trig identities (ax + b) 1 sin(ax + b) Applications of integration include all those in unit P1 cos(ax + b) sec2 (ax + b) Use of sketch graphs to identify over/under estimation Trapezium rule Numerical solution of equations Locate root graphically or by sign change Carry out iteration x n + 1 = F(x n ) Idea of sequence of approximations which converge to a root of an equation, and notation for this Understand relation between iterative formula and equation being solved Find approximate roots to a given degree of accuracy Cambridge International AS and A Level Mathematics

22 Pure Mathematics 3: Unit P3 Topic Skill Knowledge Application Checklist Assumed knowledge Content of unit P1 is assumed, and may be required in solving problems on P3 topics Algebra Solve modulus equations and inequalities, including use of: a = b a 2 = b 2 Meaning of x Meaning of quotient and remainder Use of theorems in finding factors, solving polynomial equations, finding unknown coefficients etc x a < b a b < x < a + b Carry out algebraic polynomial division Find partial fractions Factor and remainder theorems Know appropriate forms of partial fractions for denominators: (ax + b)(cx + d)(ex + f) (ax + b)(cx + d) 2 Use of first few terms, e.g. for approximations Dealing with (a + bx) n (ax + b)(x 2 + c 2 ) Expansion of (1 + x) n for rational n and x < 1 Logarithmic and exponential functions Solve equations of form a x = b and corresponding inequalities Relationship between logarithms and indices Laws of logarithms Use of laws, e.g. in solving equations Use graphs of e x and ln x Definition and properties of e x and ln x Transformation to linear form, and use of gradient and intercept 20 Cambridge International AS and A Level Mathematics 9709

23 Pure Mathematics 3: Unit P3 Topic Skill Knowledge Application Checklist Trigonometry The sec, cosec and cot functions and their relationship to cos, sin and tan Use properties and graphs of all six trig functions for angles of any magnitude Identities: sec 2 θ = 1 + tan 2 θ cosec 2 θ = 1 + cot 2 θ Expansions of: sin(a ± B) cos(a ± B) Use of these in evaluating and simplifying expressions, and in solving equations, including expressing a sin θ + b cos θ in the forms R sin(θ ± α) and R cos(θ ± α) tan(a ± B) Formulae for: sin 2A cos 2A tan 2A Differentiation Differentiate e x and ln x, sin x, cos x and tan x, together with constant multiples, sums, differences and composites Applications of differentiation include all those in unit P1 Differentiate products and quotients Parametric differentiation Implicit differentiation Cambridge International AS and A Level Mathematics

24 Pure Mathematics 3: Unit P3 Topic Skill Knowledge Application Checklist Integration Integrate: eax + b Carry out integrations using: appropriate trig identities (ax + b) 1 partial fractions sin(ax + b) cos(ax + b) Applications of integration include all those in unit P1 sec2 (ax + b) Integrate by means of decomposition into partial fractions Use of sketch graphs to identify over/under estimation kf l ( x) Recognise and integrate f ( x ) Integration by parts Integration by substitution Trapezium rule Numerical solution of equations Locate root graphically or by sign change Carry out iteration x n + 1 = F(x n ) Idea of sequence of approximations which converge to a root of an equation, and notation for this Understand relation between iterative formula and equation being solved Find approximate roots to a given degree of accuracy 22 Cambridge International AS and A Level Mathematics 9709

25 Pure Mathematics 3: Unit P3 Topic Skill Knowledge Application Checklist Vectors Find: the angle between two lines Understand r = a + tb as the equation of a straight line Determine whether two lines are parallel, intersect or are skew the point of intersection of two lines when it exists the vector equation of a line the perpendicular distance from a point to a line Find: the equation of a plane the angle between two planes the perpendicular distance from a point to a plane Understand ax + by + cz = d and (r a).n = 0 as the equation of a plane Use equations of lines and planes to solve problems, including: finding the angle between a line and a plane determining whether a line lies in a plane, is parallel to it, or intersects it finding the point of intersection of a line and a plane when it exists finding the line of intersection of two non-parallel planes Differential equations Form a differential equation from information about a rate of change General and particular solutions Use initial conditions and interpret solutions in context Solve a first order differential equation by separating variables Complex numbers Add, subtract, multiply and divide two complex numbers in cartesian form Multiply and divide two complex numbers in polar form Find the two square roots of a complex number Meaning of terms: real and imaginary parts modulus and argument conjugate cartesian and polar forms Argand diagram Equality of complex numbers Roots in conjugate pairs Geometrical interpretation of conjugation, addition, subtraction, multiplication and division Illustrate equations and inequalities as loci in an Argand diagram Cambridge International AS and A Level Mathematics

26 Mechanics 1: Unit M1 Topic Skill Knowledge Application Checklist Trigonometry knowledge required sin(90 θ) cos θ cos(90 θ) sin θ sin tan θ cos i i sin 2 θ + cos 2 θ 1 Forces and equilibrium Identify forces acting Find and use components and resultants Equilibrium of forces Use smooth contact model Forces as vectors Meaning of terms: contact force normal component frictional component limiting friction limiting equilibrium coefficient of friction smooth contact Use conditions for equilibrium in problems involving finding forces, angles, etc. Limitations of smooth contact model Use F = μr and F < μr as appropriate in solving problems involving friction Use this law in solving problems Newton s third law 24 Cambridge International AS and A Level Mathematics 9709

27 Mechanics 1: Unit M1 Topic Skill Knowledge Application Checklist Kinematics of motion in a straight line Sketch displacement-time and velocity-time graphs Distance and speed as scalars, and displacement, velocity, acceleration as vectors (in one dimension) Use of positive and negative directions for displacement, velocity and acceleration Velocity as rate of change of displacement, and acceleration as rate of change of velocity Formulae for motion with constant acceleration Interpret graphs and use in solving problems the facts that: area under v-t graph represents displacement gradient of s-t graph represents velocity gradient of v-t graph represents acceleration Use differentiation and integration to solve problems involving displacement, velocity and acceleration Use standard SUVAT formulae in problems involving motion in a straight line with constant acceleration Cambridge International AS and A Level Mathematics

28 Mechanics 1: Unit M1 Topic Skill Knowledge Application Checklist Newton s laws of motion Newton s second law Meaning of terms: mass weight Use Newton s laws in problems involving motion in a straight line under constant forces Use the relationship between mass and weight Solve constant acceleration problems involving weight: particle moving vertically particle moving on an inclined plane two particles connected by a light string passing over a smooth pulley Energy, work and power Calculate work done by constant force Concepts of gravitational potential energy and kinetic energy, and formulae mgh and ½mv 2 Power as rate of working P = Fv Solve problems involving the work-energy principle, and use conservation of energy where appropriate Use the relationship between power, force and velocity with Newton s second law to solve problems about acceleration etc. 26 Cambridge International AS and A Level Mathematics 9709

29 Mechanics 2: Unit M2 Topic Skill KnowledgwA Application Checklist Assumed knowledge Content of unit M1 is assumed, and may be required in solving problems on M2 topics Motion of a projectile Use horizontal and vertical equations of motion Constant acceleration model Limitations of constant acceleration model Derive the cartesian equation of the trajectory Solve problems, e.g. finding: the velocity at a given point or instant the range the greatest height Use the trajectory equation to solve problems, including finding the initial velocity and angle of projection Equilibrium of a rigid body Locate centre of mass of a single uniform body: using symmetry Concept of centre of mass Conditions for equilibrium Solve equilibrium problems, including toppling/sliding using data from formula list Calculate the moment of a force Locate centre of mass of a composite body using moments Uniform motion in a circle Concept of angular speed v = rω Acceleration is towards centre Use of ω 2 r or v 2 /r as appropriate in solving problems about a particle moving with constant speed in a horizontal circle Cambridge International AS and A Level Mathematics

30 Mechanics 2: Unit M2 Topic Skill KnowledgwA Application Checklist Hooke s law Meaning of terms: modulus of elasticity elastic potential energy Use of T = mx and E l 2 = m in solving 2 problems involving elastic string or springs, including those where considerations of work and energy are needed l x Linear motion under a variable force d v d v Use or v as appropriate in d t d x applying Newton s second law v d d x t = and a d d v t d d = = v x v Set up and solve differential equations for problems where a particle moves under the action of variable forces 28 Cambridge International AS and A Level Mathematics 9709

31 Probability and Statistics 1: Unit S1 Topic Skill Knowledge Application Checklist Representation of data Select suitable ways of presenting raw data Construct and interpret: stem and leaf diagram box & whisker plots Measures of central tendency: mean, median, mode Measures of variation: range interquartile range Discuss advantages and/ or disadvantages of particular representations Calculate and use these measures, e.g. to compare and contrast data histograms cumulative frequency graphs standard deviation Solve problems involving means and standard deviations Estimate median and quartiles from cumulative frequency graph Calculate mean and standard deviation using: individual data items grouped data given totals Σx and Σx 2 given Σ(x a) and Σ(x a) 2 Permutations and combinations Meaning of terms: permutation combination selection Solve problems involving selections Solve problems about arrangements in a line including those with: repetition arrangement restriction Cambridge International AS and A Level Mathematics

32 Probability and Statistics 1: Unit S1 Topic Skill Knowledge Application Checklist Probability Evaluate probabilities by: counting events in the sample space calculation using permutations and combinations Meaning of terms: exclusive events independent events conditional probability Solve problems involving conditional probabilities, e.g. using tree diagrams Use addition and multiplication of probabilities as appropriate Calculate conditional probabilities Discrete random variables Construct a probability distribution table Calculate E(X) and Var(X) Notation B(n, p) Recognise situations where the binomial distribution is a suitable model, and solve problems involving binomial probabilities For the binomial distribution: calculate probabilities use formulae np and npq The normal distribution Use normal distribution tables Idea of continuous random variable, general shape of normal curve and notation N(µ, σ 2 ) Condition for B(n, p) to be approximated by N(np, npq) Use of the normal distribution as a model Solve problems involving a normal distribution, including: finding probabilities finding µ and/or σ Solve problems involving the use of a normal distribution, with continuity correction, to approximate a binomial distribution 30 Cambridge International AS and A Level Mathematics 9709

33 Probability and Statistics 2: Unit S2 Topic Skill Knowledge Application Checklist Assumed knowledge Content of unit S1 is assumed, and may be required in solving problems on S2 topics The Poisson distribution Calculate Poisson probabilities Notation Po(µ) Mean and variance of Po(µ) Use the Poisson distribution as a model, and solve problems involving Poisson probabilities Conditions for random events to have a Poisson distribution Conditions for B(n, p) to be approximated by Po(np) Conditions for Po(µ) to be approximated by N(µ, µ) Solve problems which involve approximating binomial probabilities using a Poisson distribution and/or approximating Poisson probabilities using a normal distribution with continuity correction Cambridge International AS and A Level Mathematics

34 Probability and Statistics 2: Unit S2 Topic Skill Knowledge Application Checklist Linear combinations of random variables For a random variable X: E(aX + b) = ae(x) + b Var(aX + b) = a 2 Var(X) For random variables X and Y: E(aX + by ) = ae(x) + be(y) Solve problems using results about combinations of random variables For independent X and Y: Var(aX + by ) = a 2 Var(X) + b 2 Var(Y) For a normal variable X: ax + b has a normal distribution For independent normal variables X and Y: ax + by has a normal distribution For independent Poisson variables X and Y: X + Y has a Poisson distribution Continuous random variables Use a density function to find: probabilities the mean the variance Understand the concept of a continuous random variable. Properties of a density function: always non-negative Solve problems using density function properties total area is 1 32 Cambridge International AS and A Level Mathematics 9709

35 Probability and Statistics 2: Unit S2 Topic Skill Knowledge Application Checklist Sampling and estimation Calculate unbiased estimates of population mean and variance Meaning of terms: sample Explain why a given sampling method may be unsatisfactory Determine a confidence interval for a population mean (normal population or large sample) Determine a confidence interval for a population proportion (large sample) population Need for randomness in sampling Use of random numbers for sampling Idea of sample mean as a random variable Solve problems involving the use of 2 X + N( nv, / n), including appropriate use of the Central Limit theorem Interpret confidence intervals and solve problems involving confidence intervals Distribution of sample mean from a normal population, and for a large sample from any population Hypothesis tests Formulate hypotheses for a test Carry out a test of the value of p in a binomial distribution, using either binomial probabilities directly or a normal approximation, as appropriate Concept of a hypothesis test Meaning of terms: one-tail and two-tail null and alternative hypotheses significance level Relate the results of tests to the context and interpret Type I and Type II error probabilities Carry out a test of the mean of a Poisson distribution, using either Poisson probabilities directly or a normal approximation, as appropriate Carry out a test of a population mean where: the population is normal with known variance rejection (or critical) region acceptance region test statistic Type I and Type II errors the sample size is large Calculate probabilities of Type I and Type II errors in the above tests Cambridge International AS and A Level Mathematics

36 Section 5: Useful websites Section 5: Useful websites The websites listed below are useful resources to help you study for your Cambridge International AS and A Level Mathematics. The sites are not designed specifically for the 9709 syllabus, but the content is generally of the appropriate standard and is mostly suitable for your course. This site covers some Pure Maths and Statistics topics, but not Mechanics. Revision material is arranged by topics, and includes explanations, revision summaries and exam-style questions with answers. This site has video tutorials on topics in Pure Maths, Mechanics and Statistics. There are also other resources, including examination questions (from UK syllabuses) with videos of worked solutions. This site has revision material and test questions covering some Pure Maths topics only. This site contains resources, projects, and publications, and is intended for both learners and teachers. The main items relevant to AS and A Level revision are the course materials and the interactive A-level Audits, which you can access from the Resources option; these both cover Pure Maths, Mechanics and Statistics. The course materials are textbook style notes on selected topics, and can be found by following the Mathematics Enhancement Programme (MEP) link. The audits are online tests containing questions of progressive difficulty, and can be found by following the Test and Audits link. 34 Cambridge International AS and A Level Mathematics 9709

37 Cambridge International Examinations 1 Hills Road, Cambridge, CB1 2EU, United Kingdom tel: fax: [email protected]